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885
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THE STABILITY OF THE CRITICAL POINTS OF THE GENERALIZED GAUSE TYPE PREDATOR-PREY FISHERY MODELS WITH PROPORTIONAL HARVESTING AND TIME
DELAY
Wan Natasha Wan Hussin1*, Rohana Embong2 and Che Noorlia Noor3
1,2,3Department of Mathematics, Faculty of Computer and Mathematical Sciences,
Universiti Teknologi MARA (UiTM), 40450 Shah Alam, Selangor, Malaysia.
1*[email protected], 2[email protected],
ABSTRACT
In the marine ecosystem, the time delay or lag may occur in the predator response function, which measures the rate of capture of prey by a predator. This is because, when the growth of the prey population is null at the time delay period, the predator’s growth is affected by its population and prey population densities only after the time delay period. Therefore, the generalized Gause type predator-prey fishery models with a selective proportional harvesting rate of fish and time lag in the Holling type II predator response function are proposed to simulate and solve the population dynamical problem. From the mathematical analysis of the models, a certain dimension of time delays in the predator response or reaction function can change originally stable non-trivial critical points to unstable ones. This is due to the existence of the Hopf bifurcation that measures the critical values of the time lag, which will affect the stabilities of the non-trivial critical points of the models. Therefore, the effects of increasing and decreasing the values of selective proportional harvesting rate terms of prey and predator on the stabilities of the non-trivial critical points of the fishery models were analysed. Results have shown that, by increasing the values of the total proportion of prey and predator harvesting denoted by qx Ex and qy Ey respectively, within the range 0.3102 ≤ qx Ex ≤ 0.9984 and 0.5049 ≤ qy Ey ≤ 0.5363, the originally unstable non-trivial critical points of the fishery models can be stable.
Keywords: Fishery, Holling type II Predator Response Function, Hopf Bifurcation, Predator- prey, Selective Proportional Harvesting and Time Delay.
Received for review: 01-07-2021; Accepted: 09-08-2021; Published: 21-09-2021
1. Introduction
All living organisms like plants, animals and microorganisms and nonliving organisms like air, soil and water have complex relationships between them in the ecosystem. The relationship between prey and its predator both fish species is known as the predator-prey relationship.
Predators are dependent on prey for their continued survival. To avoid a single species from being dominant, the predator is the leading role in maintaining the prey population and improving the biodiversity within the ecosystem. Mathematical models can be applied to simulate the predator-prey population dynamics of the fish species. One of the models to analyse the predator-prey interaction is the generalized Gause type predator-prey model which
886
is based on the Logistic growth model and predator response function. A study by Hwang (1999) used a unique limit cycle for Gause type predator-prey models for systems of nonlinear ordinary differential equations since the limit cycle does not exist in the linear ordinary systems.
According to Feller (1940), almost all populations that increase asymptotically will follow the Logistic growth law to some degree. For the predator-prey relationship, the response functions between prey and predator act as the main indicator of the survival of both populations in an ecosystem. Three functional responses of the predator are Holling type I, Holling type II and Holling type III (Holling, 1965).
The Holling type I predator reaction function is a linear functional response that gives a directly proportional relationship between the rate of consumption of an individual predator and the density of its prey. Besides, the Holling type II predator reaction function is also called a hyperbolic or Michaelis-Menten functional response is used to analyse the behaviour of the predator population towards its different prey densities. The Holling type III or sigmoidal- shaped functional response was originally proposed as a model to describe the transformation of the predator’s preference from primary prey to alternative prey when the primary prey is at low densities (Kar & Matsuda, 2007). Safaa et al. (2016) analysed that the Holling type IV predator reaction function, also known as the Monod-Haldane functional response, is used to describe a condition in which the predator’s per capita rate of predation decreases at sufficiently high prey densities.
Jha & Ghorai (2017) have considered predator-prey models with Holling type I and II predation response functions with selective constant harvesting in the predator-prey systems.
Meanwhile, Martin & Ruan (2001) analysed the effects of constant-rate harvesting of prey populations and time lag on the dynamics of a predator-prey system in predation response function. Sinha & Chanda (2014) revealed that the time delay is considered in the predator- prey system because both predator and prey need their own time to grow from birth to maturity and from maturity to death. Moghadas et al. (2004) studied a non-standard numerical scheme for a generalized Gause type predator-prey model. They used the Holling type II predator reaction function f(x), denoted by:
(1) where 𝑚 > 0 is the capturing rate of prey by a predator, 𝑎 is the half saturation constant for predator population, which is the prey density at which the functional response of predator is half maximal and x is the number of the prey population. Most researchers (Jha & Ghorai, 2017; Martin & Ruan, 2001) used Holling type II predator response function with selective constant harvesting. However, this research applies selective proportional harvesting. A study by Hntsa & Mengesha (2016) indicated that to prevent the extinction of renewable resources, researchers must devise efficient harvesting strategies that produce optimum yield while maintaining renewable resources above a sustainable level. For fishery management and many other harvesting situations, it is unrealistic to consider that the harvest rate is constant which does not depend on the population, because the more fish in an area, the more will be caught.
This article begins with the determination of the first critical values of the time delay and is followed by the estimation of the critical values of the total proportion of prey and predator harvesting that can affect the stabilities of non-trivial critical points for two fishery models.
2. Non-Trivial Critical Points and Characteristic Equations of Fishery Models In this section, the derivation of the non-trivial critical points which are corresponding to both prey and predator presence in an ecosystem and characteristic equations of fishery models I and II are considered. The critical points can also be called equilibrium or fixed points, which can be analysed as steady-state solutions of a dynamical system. The steady-state solutions have
x a x mx
= + ) f(
887
the properties of stabilities. According to Zill & Cullen (2009) from the characteristic equation, the type of stability for the critical point can be determined.
2.1 Non-Trivial Critical Point and Characteristic Equation of Fishery Model I The dynamical system for fishery model I is also known as the generalized Gause type predator- prey fishery model with prey proportional harvesting and the time lag in the Holling type II predator reaction function is presented by:
(2) where,
is the number of preys in tonnes is the number of predators in tonnes is the growth rate of prey population is carrying capacity of the prey population is the capturing rate of prey by a predator
is half saturation constant for the predator population is the catchability coefficient for prey population is the effort harvesting rate of prey population is a death rate of the predator population is the time delay in months
is time in months
c is the rate of conversion of consumed prey to predator
To derive the non-trivial critical point (x*, y* ), and in Equation (2) must be zero. Therefore, the formula for x* and y* are:
(3) (4) By substituting the value of parameters from Kar (2003) which are:
the critical or fixed point obtained is (50.0750, 54.2963).
According to Martin & Ruan (2001) and Zill & Cullen (2009), and . Then the linearized system becomes:
(5)
From Equation (5), the characteristic equation (Abdallah et al., 2011; Martin & Ruan, 2001;
Ruan, 2009; Xia et al., 2009) is:
( ) ( )
( )
ïþïý ü -
+ -
+ -
=
- +
- -
=
)) ( /(
) ( ) ( )
( /
) (
) ( ))
( /(
) ( ) ( /
) ( 1 ) ( /
) (
t t y t a x t t
cmx t
wy dt t dy
t x E q t x a t y t mx k t x t rx dt t
dx x x
xy r k m a
qx
Ex
tw t
dt t
dx( )/ dy(t)/dt
) /(
cm w
aw x* = -
(
*(1 * / ) *) (
/ * /( *))
* = rx -x k -q E x mx a+x
y x x
25, y(0) and 100 ) 0 ( , 1 , 7 . 0 , 5 . 0 , 224 . 0 ,
03 . 20 , 100 ,
2 = = = = = = = =
= k a q E w c m x
r x x
x* x
X = - Y = y-y*
( )
( ) ( ) ( )
( )
ïþïý ü
*
*
*
*
*
*
*
+ -
=
+ -
+ -
- - +
=
2 2 2
*
) /(
) ( /
) /(
) ( )
( / ) ( )
2 ( ) ( /
x a t
X a cmy dt Y d
x a t Y mx x
a k t X akmy E
kq rx kr x a dt X
d x x
t
888
(6)
where,
and . By
substituting all the parameter values from Kar (2003), the resulted value is 0.4483 and is 0.1106.
2.2 Non-Trivial Critical Point and Characteristic Equation of Fishery Model II The dynamical system for fishery model II is also known as the generalized Gause type predator-prey fishery model with predator proportional harvesting and the time lag in Holling type II predator reaction function is presented by:
(7)
where is the catchability coefficient of the predator and is the effort harvesting rate of the predator. The non-trivial critical point (x*, y*) for fishery model II is calculated by letting both and in Equation (7) with zero and resulted in the coordinates of the critical values:
(8) (9) By substituting the value of parameters r = 1.8, k =100, a = 10, ,
, from Kar (2003) and Kar & Matsuda (2006), the non-trivial critical point obtained is (71.4286, 22.0408). All parameter values are taken from Kar (2003) and Kar & Matsuda (2006) because these values fit the fishery models and are in the permissible range.
Again by letting and (Martin & Ruan (2001); Zill & Cullen (2009)), the linearized system becomes:
(10)
From Equation (10), the characteristic equation (Abdallah et al., 2011; Martin & Ruan, 2001;
Ruan, 2009; Xia et al., 2009) is:
2 + l+ -lt =0
l px rxe
x =
p
(
-(a+x*)2(kr-2rx*-kqxEx)+akmy*) (
/ k(a+x*)2)
rx =(cm2x*y*a)/(a+x*)3px rx
( ) ( )
( )
ïï þ ïïý ü -
- + -
+ -
=
+ -
-
=
) ( ))
( /(
) ( ) ( )
( /
) (
)) ( /(
) ( ) ( /
) ( 1 ) ( /
) (
t y E q t
x a t y t cmx t
wy dt t dy
t x a t y t mx k t x t rx dt t dx
y
t y
t
qy Ey
dt t
dx( )/ dy(t)/dt
) /(
)
* (
y y y
yE cm w q E
aq aw
x = + - -
(
kr a x rx a x) ( )
kmy* = ( + *)- *( + *) /
5 .
=0
y yE q ,
9 . 1 , 6 . 0 , 5 .
0 = =
= c m
w x(0)=25 y( 0)=25
x*
x
X = - Y = y-y*
( ) ( )
( )
ïþ ï ý ü
*
*
*
*
*
*
*
*
+ -
=
+ -
+ -
- +
=
2 2 2
) /(
) ( /
) /(
) ( )
( ) ( / )
2 ( ) ( /
x a t
X a cmy dt Y d
x a t Y mx t X x a k akmy rx
kr x a dt X d
t
889
(11)
where and .
Substituting the parameter values resulted to = 0.8346 and = 0.0632.
3. Fishery Model I with Time Delay
For , is the root of the characteristic equation of Equation (6). Letting into Equation (6) produces:
(12)
Applying Euler’s formula, the real and imaginary parts of
the complex number in Equation (12) are:
(13)
Squaring and adding both sides of the real and imaginary equations in Equation (13) resulted to:
(14)
is obtained from the data (Kar, 2003) resulted to . The value is found by solving Equation (13) for τ. So, by multiplying and into real and imaginary parts of Equation (13) respectively, is:
(15) Considering only the first cycle of the arctangent and substituting all parameter values resulted to . All the calculated values satisfy the following Lemma 3.1.
Lemma 3.1: (Martin & Ruan, 2001)
If > , > and , then the characteristic equation for fishery model I in Equation (6) has a pair of purely imaginary roots, .
Then, to see if a Hopf bifurcation occurs or not, the transversality condition below needs to be verified
> (16)
Taking the implicit differentiation on Equation (6) with respect to produces
(17)
2+ l+ -lt =0 l py rye
(
-( + *)2( -2 *)+ *) (
/ ( + *)2)
= a x kr rx akmy k a x
py ry =cm2x*y*a/(a+x*)3
py ry
¹0
t l =iw l =iw
2 + + =0
-w pxiw rxe-iwt
) sin(
)
) cos(
(-wt = -wt +i -wt
ei
þý ü
= -
= +
0 ) sin(
: part Imaginary
0 ) cos(
- : part Real
2
wt w
wt w
x x
x
r p
r
(
4 4 2)
21 /22
2 ÷÷ø
çç ö è
æ- ± +
± = px px rx
w
0489 .
2 =0
w+ l =±0.2211i t0+
)
sin(wt cos(wt)
+
tM
(
1/)
tan 1(
/) (
+ 2 /)
, =0,1 ,2 ,...= + - + +
+ px M M
M w w p w
t
0321 .
0+ =5 t
px 0 rx 0 t =t0+
±i
w
+( )
0 /0 t t
l d
Re
d + + 0
t
(
l)
l(
t l)
t
l/ ) (2 ) lt / /
(d d -1 = + px e rx -
890
According to Martin & Ruan (2001), the formula to prove the transversality condition is
(18) where is the real part of the complex number and is the sign of the real number whether positive or negative. By substituting Equation (17) into Equation (18) we obtain
(19)
Substituting all parameter values in Equation (19), we get
(20)
Since the value obtained is positive in Equation (20), Theorem 3.1 below is satisfied.
Theorem 3.1: (Martin & Ruan, 2001)
> , > and , then the critical point (x*, y* ) of predator-prey fishery model I has a pair of purely imaginary roots for . The critical point of fishery model I is stable for ˂ and unstable for ˃ . The system undergoes a Hopf bifurcation at .
4. Fishery Model II with Time Delay
For , is the root of Equation (11). The process to determine the eigenvalue,
for fishery model II is the same as in fishery model I. Thus, Equation (14) can be written as Equation (21) in the fishery model II.
(21)
Substituting parameter values from Kar (2003), we obtained . Meanwhile from Equation (15), by considering the value of = 0.8346. Both Lemma 3.1 and Theorem 3.1 hold for fishery model II.
5. Numerical Simulation
Numerical methods such as the Runge-Kutta first order (RK1), Runge-Kutta third order (RK3), Runge-Kutta fourth order (RK4) and Runge-Kutta-Fehlberg Fourth-Fifth order (RKF 45) have been used to solve the system of differential equations. Mullen (2015) applied and compared numerical methods RK1 and RK4 to solve the Optimal Velocity Model for one- and two-car systems. Meanwhile, the RK3 method was implemented by Rini & Alyauma (2018) to solve the Nth-Order Fuzzy Differential Equations based on the Combination of Arithmetic, Harmonics and Geometrics Means with initial values. Furthermore, Paul et al. (2016) applied
{ d Re l d t }
l iwsign { Re d l d t }
l iwsign ( ) /
== ( / )
-1 =Re sign
{
( )/}
{
( 2 2 2)/ 2}
d Re d sign px rx
sign l t l=iw = w +
{ d ( Re ) / d } sign {
24.4248}
sign l t
l=iw=
px 0 rx 0
t
=t
0+±i
w
+t
=t
0+t t0+ t t0+ t0+
¹0
t l =iw l =iw
(
4 4 2)
21 /22
2 ÷÷ø
çç ö è
æ- ± +
± = py py ry
w
i 0755 .
±0 l = 6103
.
0+ =19
t py
891
the RKF 45 method and the Laplace Adomian Decomposition method (LADM) to solve the Lotka Volterra prey-predator model. The results showed that RKF 45 was a suitable and accurate tool for solving the mentioned model. Hence, this research applied RKF 45 based on Mathews & Fink (2004) on the visualization of the stabilities of the non-trivial critical points of the two fishery models. By using this numerical method, the graphs of the predator and prey versus time and the phase portrait of prey and predator population concerning time can be plotted.
5.1 Effect of Time Delay on the Stability of the Non-Trivial Critical Point of Fishery Model I
In this subsection, we verify all mathematical results and τ = 5.0321 obtained in section 3 on the stability of the non-trivial critical point of fishery model I at time delays equal to 4.00, 5.0321 and 6.00.
Figure 1(a). Predator and prey population versus
time for fishery model I at Figure 1(b). Phase portrait of fishery model I at
Based on figure 1(a), periodic oscillations occur on both prey and predator populations at 20 to 175 months. After the 175th month, both prey and predator populations converge to their equilibrium population, which are 50.0750 and 54.2963 tonnes of fishes respectively. In figure 1(b) both prey and predator populations spiral toward their critical values as time increases. At τ = 4.00 the non-trivial equilibrium point (50.0750,54.2963) will be an asymptotically stable focus.
Figure 2(a). Predator and prey population versus time for fishery model I at
Figure 2(b). Phase portrait of fishery model I at i
2211 .
±0 l =
. 00 .
=4
t t =4.00.
. 0321 .
=5
t t =5.0321.
892
In figures 2(a) and 2(b), both prey and predator populations oscillate about their critical point or in other words, the prey and predator population neither approach nor recede away from the non-trivial critical point (50.0750, 54.2963). The eigenvalues obtained from the characteristic equation in Equation (6) at τ = 5.0321 are . The Hopf bifurcation, which is the visual aspect of the periodic orbit through a local change in the stability properties of a critical point as shown in figures 2(a) and 2(b). The time delay, τ = 5.0321 in fishery model I, indicates the transition from the steady condition of prey and predator population in the marine ecosystem to periodic conditions. Research conducted by Baisad & Moonchai (2018), found that a Hopf bifurcation occurs when a system of differential equations has a pair of complex conjugate eigenvalues of the Jacobian matrix at the fixed point.
Figure 3(a). Predator and prey population versus
time for fishery model I at . Figure 3(b). Phase portrait of fishery model I at From the numerical simulation, the limit cycle, which is the periodic solution or orbit, occurs at τ = 6.00, which means that the bifurcation occurs before τ = 6.00. From figures 3(a) and 3(b), there is a bifurcating periodic solution, which causes the non-trivial critical point (50.0750, 54.2963) of the fishery model I to become unstable because both prey and predator populations do not converge to their critical number in finite time (600 months). Figure 3(a) shows the large periodic oscillations on both prey and predator populations around the critical point (50.0750, 54.2963). This indicates that the mentioned critical point is unstable. At this level of time delay, the growth of the predator population is affected by its population and prey population densities that cause the critical point of the model to be unstable.
5.2 Effect of Time Delay on the Stability of the Non-Trivial Critical Point of Fishery Model II
In this subsection, we verify all mathematical results and τ = 19.6103 obtained in section 4 regarding the stability of the non-trivial fixed point for fishery model II at time delays equal to 15.00, 19.6103 and 21.00.
i 2211 .
±0
= l
00 .
=6
t t =6.00.
i 0755 .
±0 l =
893 Figure 4(a). Predator and prey population versus time for fishery model II at .
Figure 4(b). Phase portrait of fishery model II at
Figure 4(b) shows that the fixed point is an asymptotically stable focus because both prey and predator populations approach the fixed point (71.4286, 22.0408). In figure 4(a), there is an occurrence of small periodic oscillation for both prey and predator populations from 125 to 375 months. After the 375th month, both prey and predator populations converge to the equilibrium number of populations, 71.4286 and 22.0408 tonnes of fishes respectively.
Figure 5(a). Predator and prey population versus
time for fishery model II at . Figure 5(b). Phase portrait of fishery model II at In figure 5(a), both prey and predator populations reach the periodic oscillations that are induced by Hopf bifurcation around the non-trivial critical point up to 800 months. Figure 5(b) shows that both prey and predator populations oscillate about the equilibrium point (71.4286, 22.0408). Prey and predator population neither approach nor recede away from the non-trivial critical point (71.4286, 22.0408). At τ = 19.6103, the roots from the characteristic equation in Equation (11) are Based on the eigenvalue analysis of the nonlinear dynamical system, when the eigenvalues, are pure imaginary numbers, the type of stability of the critical point can either be the stable centre, unstable spiral point or asymptotically stable spiral point in the nonlinear system (Zill & Cullen, 2009).
00 .
=15
t t =15.00.
6103 .
=19
t t =19.6103.
.
0755 .
0 i
±
=
l l
894 Figure 6(a). Predator and prey population versus
time for fishery model II at Figure 6(b). Phase portrait of fishery model II at Figure 6(a) shows the bifurcating periodic solution produces a non-stable equilibrium point at point (71.4286, 22.0408). Thus, both prey and predator populations do not converge at the critical point (71.4286, 22.0408). Figures 6(a) and 6(b) show the limit cycle that describes the occurrence of oscillatory behaviour by the Hopf bifurcation on both prey and predator populations, hence the non-stable critical point does not reach its equilibrium number of populations. At this level of time delay which is τ = 21.00, the growth of predator and prey population is unstable.
5.3 Effect of Total Proportion of Prey Harvesting ( ) on the Stability of the Non-Trivial Critical Point of Fishery Model I
In this subsection, we analyse the critical values of the total proportion of prey harvesting on the fishery model I when τ = 6.00, which can stabilise the non-trivial critical point that is unstable as shown in figure 3(a). From numerical testing, the values qx Ex within the range [0.3102, 0.9984], changes an originally unstable non-trivial critical point to become stable.
Figure 7. Predator and prey population versus time
for fishery model I with qx Ex = 0.3102. Figure 8. Predator and prey population versus time for fishery model I with qx Ex = 0.9984.
. 00 .
=21
t t =21.00.
x xE q
895
At , the originally unstable non-trivial critical point become stable as shown in figure 7. This is because the equilibrium number of predator population y* from Equation (4) decreases while the equilibrium number of prey population x* remains constant (x* = 50.0750). There is an oscillation or bifurcation that occurs on both prey and predator populations before the 500th month. As time increases, both prey and predator populations converge to the critical point (x*, y* ) = (50.0750, 48.2533). From figure 8,
both prey and predator populations converge to the critical point (50.0750, 0.0072). The prey population is dominant in the marine ecosystem, while the predator population is driven to extinction. According to Dawed et al. (2014), this situation is preferable because the prey population does not receive any threat from the predator population to survive in the marine ecosystem. Thus, the prey population would grow at a natural rate. The competition that is only present is in between the prey population itself, which is hunting for the same food when the predator population is absent in the marine ecosystem. Beyond the critical proportional harvesting rate of prey the number of prey populations will be decreased.
5.4 Effect of Total Proportion of Predator Harvesting ( ) on the Stability of the Non-Trivial Critical Point of Fishery Model II
In this subsection, we analyse the critical values of the total proportion of predator harvesting on the fishery model II with τ = 21.00 which can stabilise the non-trivial fixed point that is unstable as shown in figure 6(a). From the numerical simulation, the range value of the parameter qy Ey that can stabilise the unstable non-trivial critical point is [0.5049, 0.5363].
Figure 9. Predator and prey population versus time for fishery model II with
Figure 10. Predator and prey population versus time for fishery model II with From figure 9, the non-trivial critical point (x*, y* ) = (74.3819, 20.4793) of fishery model II for τ = 21.00 and qy Ey = 0.5049 is stable, which is described by the small periodic oscillations between 200 to 700 months. After the 700th month, both prey and predator populations will converge to their fixed point (x*, y* ) = (74.3819, 20.4793) which is the critical point. For qy Ey
= 0.5363and τ = 21.00, both prey and predator populations approach the equilibrium point (99.9325, 0.0703) and stabilise at the beginning of the 25th month, as shown in figure 10. The equilibrium number of predator population approaches null, while the prey population reaches its full capacity. In figures 9 and 10 populations of prey are increasing as the total proportion of predator harvesting is increasing. This is because the more predator is proportionally harvested, the more prey population will be in the marine ecosystem. The prey population does not get any threat anymore from predator populations to survive in the marine ecosystem.
3102 .
=0
x xE q
, 9984 .
=0
x xE q
), 9985 . 0 (qxEx =
y yE q
. 5049 .
=0
y yE
q qyEy =0.5363.
896
Hence, the predator population is driven to extinction, whereas the prey population is dominant in the marine ecosystem.
6. Conclusion
From this research, we observed that the first critical values of time lag in the Holling type II predator reaction function, and in fishery model I and II respectively, enable both stable non-trivial critical points to become unstable due to the existence of Hopf bifurcation that may lead to the existence of a limit cycle. We conclude that the proportional harvesting rate of the prey and predator populations had a stabilising effect on the stabilities of the non-trivial equilibrium points of the fishery systems. Applying the parameter values from Kar (2003) and Kar & Matsuda (2006), we obtain and , the first critical values of time delay for the fishery models I and II, respectively. We derived new critical or equilibrium points (x*, y*) as written in Equation (3),(4),(8) and (9) for both models. The non- trivial critical points became stable at τ < 5.0321 and τ < 19.6103, in fishery models I and II respectively. To stabilise the non-trivial critical points of both fishery models, the total proportion of prey harvesting qx Exis in the interval [0.3102, 0.9984] and the total proportion of predator harvesting qy Eyis given by 0.5049 ≤ qy Ey ≤ 0.5363. We conclude that the non- trivial fixed points become stable when the predator population is reduced in the marine ecosystem using both fishery models I and II, inevitably enhancing the growth of the prey population.
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0321 .
=5
t t =19.6103
0321 .
=5
t t =19.6103
897
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